In one episode a series of unfortunate events turns Gomer's simple errand into an impromptu world-tour. He calls Sgt. Carter from the airport in Paris:
Gomer: (on the phone) "That's silly, Sergeant. How could I get you arrested for murder?" (beat) "Oh."
Sgt. Carter's nightmare in the episode where he sends Pyle to paratrooper school, only to learn that Gomer named him beneficiary for his life-insurance policy; "Ten thousand dollars!"
In the same episode, watching Pyle's first jump:
Carter: "That's Pyle alright; he's trying to fly."
In one episode, Sgt. Carter tried to get Gomer out of the military when it looks like his hitch is about up. So he helps him out - he finds him a good job, sets him up with a nice lady, and generally does everything possible to ensure him a comfortable post-military life. But Gomer re-enlists. His reasoning? Sgt. Carter was being so nice, he had to repay the favor and help him some more.
In the very first episode, Gomer Overcomes the Obstacle Course, Gomer enters a tiny culvert headfirst and, somehow, emerges feet-first. Sgt. Carter's reaction sells it.
Sgt. Carter's reaction to finding that Gomer has stashed a girl in his (Carter's) locker (long story). It begins with one of the most incredible double takes you'll ever see and builds from there.